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Keywords:
inequality; Hardy type inequality; Hardy operator; Riemann-Liouville operator; $q$-analysis; sharp constant; discrete Hardy type inequality
inequality; Hardy type inequality; Hardy operator; Riemann-Liouville operator; $q$-analysis; sharp constant; discrete Hardy type inequality
Summary:
Some $q$-analysis variants of Hardy type inequalities of the form $$ \int _0^b \bigg (x^{\alpha -1} \int _0^x t^{-\alpha } f(t) {\rm d}_q t \bigg )^{p} {\rm d}_q x \leq C \int _0^b f^p(t) {\rm d}_q t $$ with sharp constant $C$ are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.
References:
[1] Al-Salam, W. A.: Some fractional $q$-integrals and $q$-derivatives. Proc. Edinb. Math. Soc., II. Ser. 15 (1966), 135-140. DOI 10.1017/S0013091500011469 | MR 0218848 | Zbl 0171.10301
[2] Bangerezako, G.: Variational calculus on $q$-nonuniform lattices. J. Math. Anal. Appl. 306 (2005), 161-179. DOI 10.1016/j.jmaa.2004.12.029 | MR 2132895 | Zbl 1095.49005
[3] Bennett, G.: Factorizing the Classical Inequalities. Memoirs of the American Mathematical Society 576 AMS, Providence (1996). MR 1317938 | Zbl 0857.26009
[4] Bennett, G.: Inequalities complementary to Hardy. Q. J. Math., Oxf. II. Ser. 49 (1998), 395-432. DOI 10.1093/qmathj/49.4.395 | MR 1652236
[5] Bennett, G.: Series of positive terms. Conf. Proc. Poznań, Poland, 2003 Z. Ciesielski et al. Banach Center Publications 64 Polish Academy of Sciences, Institute of Mathematics, Warsaw (2004), 29-38. MR 2099457 | Zbl 1058.26011
[6] Bennett, G.: Sums of powers and the meaning of $l^p$. Houston J. Math. 32 (2006), 801-831. MR 2247911
[7] Cass, F. P., Kratz, W.: Nörlund and weighted mean matrices as operators on $l_p$. Rocky Mt. J. Math. 20 (1990), 59-74. DOI 10.1216/rmjm/1181073159 | MR 1057975
[8] Ernst, T.: A Comprehensive Treatment of $q$-calculus. Birkhäuser Basel (2012). MR 2976799 | Zbl 1256.33001
[9] Ernst, T.: The History of $q$-Calculus and a New Method. Uppsala University Uppsala (2000), http://www2.math.uu.se/research/pub/Ernst4.pdf
[10] Exton, H.: $q$-Hypergeometric Functions and Applications. Ellis Horwood Series in Mathematics and Its Applications Halsted Press, Chichester (1983). MR 0708496 | Zbl 0514.33001
[11] Gao, P.: A note on Hardy-type inequalities. Proc. Am. Math. Soc. 133 (2005), 1977-1984. DOI 10.1090/S0002-9939-05-07964-5 | MR 2137863 | Zbl 1068.26015
[12] Gao, P.: Hardy-type inequalities via auxiliary sequences. J. Math. Anal. Appl. 343 (2008), 48-57. DOI 10.1016/j.jmaa.2008.01.024 | MR 2409456 | Zbl 1138.26309
[13] Gao, P.: On $l^p$ norms of weighted mean matrices. Math. Z. 264 (2010), 829-848. DOI 10.1007/s00209-009-0490-2 | MR 2593296 | Zbl 1190.47012
[14] Gao, P.: On weighted mean matrices whose $l^p$ norms are determined on decreasing sequences. Math. Inequal. Appl. 14 (2011), 373-387. MR 2816127 | Zbl 1237.47008
[15] Gauchman, H.: Integral inequalities in $q$-calculus. Comput. Math. Appl. 47 (2004), 281-300. DOI 10.1016/S0898-1221(04)90025-9 | MR 2047944 | Zbl 1041.05006
[16] Hardy, G. H., Littlewood, J. E., Pólya, G.: Inequalities. (2nd ed.), Cambridge University Press Cambridge (1952). MR 0046395 | Zbl 0047.05302
[17] Jackson, F. H.: On $q$-definite integrals. Quart. J. 41 (1910), 193-203.
[18] Kac, V., Cheung, P.: Quantum Calculus. Universitext Springer, New York (2002). MR 1865777 | Zbl 0986.05001
[19] Krasniqi, V.: Erratum: Several $q$-integral inequalities. J. Math. Inequal. 5 (2011), 451. DOI 10.7153/jmi-05-39 | MR 2865561 | Zbl 1225.26046
[20] Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy Inequality. About Its History and Some Related Results. Vydavatelský Servis Plzeň (2007). MR 2351524 | Zbl 1213.42001
[21] Kufner, A., Maligranda, L., Persson, L.-E.: The prehistory of the Hardy inequality. Am. Math. Mon. 113 (2006), 715-732. DOI 10.2307/27642033 | MR 2256532 | Zbl 1153.01015
[22] Kufner, A., Persson, L.-E.: Weighted Inequalities of Hardy Type. World Scientific Singapore (2003). MR 1982932 | Zbl 1065.26018
[23] Maligranda, L.: Why Hölder's inequality should be called Rogers' inequality. Math. Inequal. Appl. 1 (1998), 69-83. DOI 10.7153/mia-01-05 | MR 1492911 | Zbl 0889.26001
[24] Miao, Y., Qi, F.: Several $q$-integral inequalities. J. Math. Inequal. 3 (2009), 115-121. DOI 10.7153/jmi-03-11 | MR 2502546 | Zbl 1181.26009
[25] Persson, L.-E., Samko, N.: What should have happened if Hardy had discovered this?. J. Inequal. Appl. (electronic only) 2012 (2012), Article ID 29, 11 pages. MR 2925639 | Zbl 1282.26038
[26] Stanković, M. S., Rajković, P. M., Marinković, S. D.: On $q$-fractional derivatives of Riemann-Liouville and Caputo type. arXiv: 0909.0387v1[math.CA], 2 Sept. 2009.
[27] Sulaiman, W. T.: New types of $q$-integral inequalities. Advances in Pure Math. 1 (2011), 77-80. DOI 10.4236/apm.2011.13017 | MR 2724033
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